# Nonverifiability: Subgame-Perfect Implementation

From proposition 6.4, a necessary condition for unique Nash implementation is that an allocation rule a(·) be monotonic. Any allocation rule that fails to be monotonic will also fail to guarantee unique Nash implementation. Then, one may wonder if refinements of the Nash equilibrium concept can still be used to ensure unique implementation. The natural refinement of subgame perfection will appear when one moves to a game with sequential moves, where the principal and the agent take turns sending messages to the court. An allocation rule a(θ) is  uniquely  implementable  in  subgame-perfect  equilibrium  by  a  mechanism  g¯(·) provided that its unique subgame-perfect equilibrium yields allocation a(θ) in any state θ.

Instead of presenting the general theory of subgame-perfect implementation, which is quite complex, we propose a simple example showing the mechanics of the procedure. Let us first single out a principal-agent setting where the first- best allocation rule is nonmonotonic. As we know from the last section, this calls for a more complex modelling of information than what we have used so far if we remain in the context of principal-agent models with quasi-linear objective functions. Consider a principal with utility function V = S(q) − t independent of the state of nature θ. For simplicity, we assume that , where µ and λ are common knowledge. The agent instead has a utility function U = , where θ = (θ1, θ2) is now a bidimensional state of nature.

The first-best outputs q1, θ2) are given by the first-order conditions S(q1, θ2)) = θ1 + θ2q1, θ2). We immediately find that .

We assume that each parameter θi belongs to . A priori, there are four possible states of nature and four first-best outputs. Assuming that i.e.,  µ − h = θ + θ¯,  we  are  left  with  three  first-best  outputs 1,  and that we assume to be all positive.

Of course, even if the  production level is the same  in states , the  agent  has  different  costs  and  should  receive  different  transfers  tˆ1*  and tˆ2*   from the principal in those two states of nature. We denote by t and t¯*  the transfers in the other states of nature.

In figure 6.7, we have represented the first-best allocations corresponding to the different states of nature.

Importantly, the indifference curve of a (θ, θ¯)-agent going through the first-best allocation C of a (θ, θ¯)-agent (dotted curve in figure 6.7) is tangent to and always  above  that  of  a  (θ¯, θ)-agent.11   This  means  that  one  cannot  find  any  allo- cation (tˆ, qˆ) such that condition (A) of definition 6.6 holds. In other words, the first-best allocation rule a(θ) is nonmonotonic in this bidimensional example. To see  more  precisely  why  it  is  so,  note  that  any  mechanism  g˜(·) implementing  the first-best  allocation  a(θ¯, θ) must  be  such  that  Figure 6.7: First-Best Allocations

But, since the indifference curve of a (θ, θ¯)-agent through C is above the one  of  a  (θ¯, θ)-agent,  this  inequality  also  implies  that Since the principal’s utility function does not depend directly on θ, the pair of strategies (ma(θ¯, θ); (m*p(θ¯, θ)) that implements the allocation remains an equilibrium in state (θ, θ¯). Hence, there is no hope of finding a unique Nash implementation of the first-best outcome.

Let us now turn to a possible unique implementation using a three-stage extensive form mechanism and the more stringent concept of subgame-perfection. The reader should be convinced that there is not too much problem in elicit-ing the preferences of the agent in states (θ, θ) and (θ¯, θ¯).   Hence, we will focus on a “reduced” extensive form that is enough to highlight the logic of subgame- perfect implementation. The objective of this extensive form is to have the agent truthfully  reveal  the  state  of  nature  when  either  (θ¯, θ) or  (θ, θ¯) occurs. Figure 6.8: Subgame-Perfect Implementation

In figure 6.8 we have represented such an extensive form.

The  mechanism  to  be  played  in  both  states  (θ, θ¯) and  (θ¯, θ) is  a  three-stage game  with  the  agent  moving  first  and  announcing  whether  (θ, θ¯) or  (θ¯, θ) has been  realized.  If  (θ, θ¯) is  announced,  the  game  ends  with  the  allocation . If  (θ¯, θ)  is  announced,  the  principal  may  agree  and  then  the  game  ends  with the  allocation or  challenge.  In  the  latter  case,  the  agent  has  to  choose between  two  possible  out-of-equilibrium  allocations .  We  have a greater flexibility with respect to Nash implementation, since now the agent has sometimes to choose between two allocations that are nonequilibrium ones instead of between an out-of-equilibrium one and an equilibrium one, as under Nash implementation. We want to use this flexibility to obtain in the state of nature (θ, θ¯) and in the state of nature (θ¯, θ). To do so, we are going to choose  the  allocations in  such  a  way  that  the  agent  prefers  a different allocation in different states of the world. Specifically, we choose them to have and Then,  since  at  stage  3  the  agent  chooses in  state  (θ¯, θ),  to  obtain the principal should not be willing to challenge the agent’s report at stage 2 of the game. This means that one should have Finally, the agent with type (θ¯, θ) should prefer to report truthfully that (θ¯, θ) has realized, i.e.: Now let us see how we can obtain in the state of nature (θ, θ¯). Since the  agent  chooses in  state  (θ, θ¯),  the  principal  should  be  willing  to  chal- lenge, i.e., Expecting this behavior by the principal, the agent should not be willing to announce  (θ¯, θ) when  the  state  of  nature  is  (θ, θ¯).  This  means  that  the  following inequality must also hold: The  remaining  question  is  whether  there  exists  a  pair  of  contracts that  satisfy  constraints  (6.21)  to  (6.26).  The  response  can  be  given graphically (figure 6.9). By definition, (resp. ) should be above (resp. below)  the  principal’s  indifference  curve  going  through  C.  Note  that  for  q > qˆ*, the indifference curves of an agent with (θ, θ¯) have a greater slope than those of an agent with type (θ¯, θ). This helps to construct very easily the out-of-equilibrium allocations ,  as  in  figure  6.9.

Remark: Subgame-perfect implementation is beautiful and attractive, but it should be noted that it has been sometimes criticized because it relies excessively on rationality. The kind of problem at hand can be illustrated with our example of figure 6.8. Indeed, when state (θ, θ¯) realizes and the principal has to decide to move at the second stage, he knows that the agent has already made a suboptimal move. Why should he still believe that the agent will behave optimally at stage 3, as needed by subgame-perfect implementation? Figure 6.9: Subgame-Perfect Implementation

Moore and Repullo (1988) present a set of conditions ensuring subgame-perfect implementation in general environments, notice-ably those with more than two agents. The construction is rather complex but close in spirit to our example. Abreu and Matsushima (1992) have developed the concept of virtual-implementation of an allocation rule. The idea is that the allocation rule may not be implemented with probability one but instead with very high probability. With this implementation concept, any allocation rule can be virtually implemented as a subgame-perfect equilibrium.

Source: Laffont Jean-Jacques, Martimort David (2002), The Theory of Incentives: The Principal-Agent Model, Princeton University Press.